However, in many cases, the ItR has to be considered as a stochastic function, instead of a deterministic one. These situations may originate from (a) an intrinsically stochastic dynamical system, e.g. stochastic differential equations modeling a physical diffusion process or stock prices [1]}; the stochastic model of climate variability including the non-average `weather' component as random forcing terms [2]}, and (b) some uncertain variables that are not easily incorporated in the input parameter space. Under such situations, the probability distribution of the response is critically influenced by the uncertainty in the ItR (in addition to the uncertainty of input parameters). Moreover, the uncertainty of the ItR (in particular the variance of response) is often inhomogeneous for different input parameters. For example, the uncertainty of power production of wind turbine is different among various wind speed inputs [3]}; the uncertain ItR of ship motion resulted from uncertain initial conditions is sensitive to the wave group parameters as the input. A sequential BED method, which considers the heteroscedastic uncertainty in ItR and its effect on extreme-event probability is not yet available.
